122 XIII INTEGRATION Necessity. Suppose that / is integrable, then there exist functions g e & and heS such that g g/<£ h and jj*(h -g)£ is. Also ((13.5.1) and (13.5.3)) there exists u e JfR(X) such that u ^ h and jj*(h - u) g |e. Since |/- z/| ^ |A - «| + \h - gl it follows by (13.5.3) that Sufficiency. If u e Jf R(X) is such that ju*(|/- w|) <; e, then by definition (13.5.5) there exists a function u e </ such that |/— w| ^ t? and ju*(y) ^ 2e. But the relation —v^f—u^v can be written in the form — v -f w ^/^ z> + w (w being finite); also we have —v + ue^ and t>+we«/ (12.7.5), and ^*(t? + w — (— 0 H- w)) = 2fi*(v) ^ 4e; hence /is integrable. (1 3.7.3) 77z<? 5-^ JS?i(X, //) (also denoted by S\f^ or j?i) o functions on X w ^z vector space over R, a«rf the mapping f\~* \fdfj, is a positive linear form on <&% (i.e., the relation f^. 0 implies \fdjj,^G). If /is finite and integrable, then clearly so is affor every scalar # eR, and we have \afdjLi = a (fdfi. If /and g are finite and integrable, it follows from (13.5.6) applied to /and g and to —/and — g that ffdfJL +\gdn£ [(/+ g) dp £ (**(/+ 0) dp £ [fdp + (gd^ which completes the proof. It follows that if /and g are integrable functions (finite or not) on X with values in R, then/+ g (which is defined almost everywhere) is integrable, and that J(/+ g) dp = jfdf* + JV d/i. (13.7.4) Iff is integrable, then so are I/I,/"1" andf~9 and we have fdfi £ I/I dp. (13.7.4.1) rff and g are integrable, then so are sup(/ g) andinf(f, g). If u e JTR(X) is such that ju*(|/— u\) ^ 6, then since | I/I -1«| | £ I/-«|,s